Problem 11 - Applied Calculus

Question

Evaluate each expression without using a calculator. $$ \left(\frac{3}{2}\right)^{-3} $$

Step-by-Step Solution

Answer

The value is \( \frac{8}{27} \).

Step 1: Understand Negative Exponent

A negative exponent indicates that the base should be taken as a reciprocal and then raised to the opposite positive power. For example, \( a^{-n} = \frac{1}{a^n} \). Here, \( a = \frac{3}{2} \) and \( n = 3 \), so this means \( \left(\frac{3}{2}\right)^{-3} = \frac{1}{\left(\frac{3}{2}\right)^3} \).

Step 2: Apply Exponentiation Property

Now, calculate \( \left(\frac{3}{2}\right)^3 \) by multiplying \( \frac{3}{2} \) by itself three times: \( \left(\frac{3}{2}\right)^3 = \frac{3}{2} \times \frac{3}{2} \times \frac{3}{2} = \frac{27}{8} \).

Step 3: Calculate Reciprocal

Since \( \left(\frac{3}{2}\right)^{-3} = \frac{1}{\left(\frac{3}{2}\right)^3} \), substitute the value computed in the previous step to find \( \frac{1}{\left(\frac{3}{2}\right)^3} = \frac{1}{\frac{27}{8}} \). This is equivalent to multiplying by the reciprocal: \( \frac{8}{27} \).

Key Concepts

Reciprocal

A reciprocal is a special mathematical concept where you essentially "flip" a fraction. This means that if you take a fraction like \( \frac{3}{2} \), its reciprocal would be \( \frac{2}{3} \).
The reciprocal of a number \( a \), is \( \frac{1}{a} \). For fractions, you simply swap the numerator (top number) and the denominator (bottom number).
Reciprocals are useful in simplifying negative exponents because a negative exponent means taking the reciprocal of the base before applying the ordinary exponentiation.

• Example: The reciprocal of \( \left(\frac{3}{2}\right)^{-3} \) is calculated as \( \frac{1}{\left(\frac{3}{2}\right)^3} \).

Understanding reciprocals can simplify many operations, particularly when working with negative exponents in expressions.

Exponentiation

Exponentiation is a fundamental operation in mathematics where a number, known as the base, is multiplied by itself a specific number of times indicated by the exponent. For instance, \( a^3 \) means \( a \times a \times a \).
In the context of the expression \( \left(\frac{3}{2}\right)^{-3} \), we first find \( \left(\frac{3}{2}\right)^3 \), which involves multiplying \( \frac{3}{2} \) by itself three times.
This results in \( \left(\frac{3}{2}\right)^3 = \frac{27}{8} \).

• The exponent indicates how many times to multiply the base by itself.
• An exponent of 3 means to multiply the base three times.
• Negative exponents involve reciprocals in combination with regular exponentiation.

Mastering exponentiation is crucial for handling various mathematical operations, including rational expressions.

Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials or algebraic expressions. They often involve variables and can be simplified, factored, or manipulated in various ways.
For the expression \( \left(\frac{3}{2}\right)^{-3} \), we see the use of rational numbers being raised to a power. Once transformed by the negative exponent rule to find the reciprocal, it simplifies to a rational expression \( \frac{8}{27} \).
Here are a few key points about rational expressions:

• They are composed of a fraction with polynomials in the numerator and denominator.
• Operations include finding reciprocals, simplifying, multiplying, and dividing.
• Understanding rational expressions is important for solving complex algebra problems.

Grasping rational expressions allows for a better understanding of how to evaluate and manipulate expressions involving variables and numbers.